RESOURCE ALLOCATION FOR OFDM SYSTEMS IN THE
PRESENCE OF TIME-VARYING CHANNELS
Mort Naraghi-Pour and Xiang Gao
Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
Keywords:
OFDM, resource allocation, time-varying channel, channel estimation, channel prediction.
Abstract:
Adaptive resource allocation schemes for OFDM systems designed assuming static channels are known to
experience significant performance loss when the channel is time-varying. The inaccuracy in channel state
information (CSI) due to outdated channel estimates has been recognized as the main reason for this problem.
To mitigate this effect, we present robust bit and power allocation schemes based on the predicted channel
state information. Simulation results show that channel prediction schemes can significantly improve the
performance of resource allocation algorithms over time-varying channels.
1 INTRODUCTION
Adaptive resource allocation has been shown to result
in significant performance improvement for OFDM
systems over frequency selective channels (Chow
et al., 1995; Pan et al., 2004; Krongold et al., 2000;
Gao and Naraghi-Pour, 2006). In these studies it is
assumed that the transmitter has complete and perfect
knowledge of the channel state information (CSI).
Then for each subcarrier a proper size of modulation
signal set and transmit power is selected according to
the channel frequency response such that the desired
quality of service (QoS) can be achieved with the
maximum spectral efficiency. In practice, however,
CSI is obtained at the receiver through channel esti-
mation and the noise in the received signal may cause
estimation errors. Furthermore, the wireless channel
is often time-varying and thus transmission and pro-
cessing delay will make the CSI estimates outdated.
Previous research (Ye et al., 2002; Leke and Cioffi,
1998; Wyglinski et al., 2004; Falahati et al., 2004)
has confirmed that the performance of most adaptive
modulation schemes assuming perfect knowledge of
CSI will degrade significantly even with moderate er-
rors in the estimated CSI. While the estimation error
can be suppressed using efficient channel estimation
techniques, for time-varying channels, the difficulties
due to outdated CSI estimates remain.
In (Eyceoz et al., 1997) it is shown that the state
of a frequency-flat fading channel can be reliably
predicted from the previous observations across a
long range of time. This motivates the prediction of
frequency-selective channels for the OFDM system
since, using OFDM, the wideband channel is trans-
formed into a number of flat-fading sub-channels. In
this paper we adopt the idea of CSI prediction and
propose to perform resource allocation based on the
predicted CSI.
The remainder of paper is organized as follows.
Some preliminary results are presented in Section 2,
in which we also motivate the need for channel pre-
diction in resource allocation over time-varying chan-
nels. In Section 3, different channel predictors using
Wiener filter or adaptive filters are discussed. Bit and
power allocation based on the predicted CSI is dis-
cussed in Section 4 and the simulation results are pre-
sented in Section 5. Finally, conclusions are drawn in
Section 6.
117
Naraghi-Pour M. and Gao X. (2007).
RESOURCE ALLOCATION FOR OFDM SYSTEMS IN THE PRESENCE OF TIME-VARYING CHANNELS.
In Proceedings of the Second International Conference on Wireless Information Networks and Systems, pages 117-124
DOI: 10.5220/0002150301170124
Copyright
c
 SciTePress

2 PRELIMINARY ANALYSIS
2.1 Channel Model
The equivalent lowpass impulse response of a time-
varying, frequency-selective multipath fading channel
can be written as
c(τ;t) =
D−1
∑
i=0
r
i
(t)δ(τ−τ
i
) (1)
where we assume that the path gains r
i
(t) are wide-
sense stationary uncorrelated processes (WSS-US)
(Proakis, 2000; Stuber, 2000). The WSS-US assump-
tion implies that
E[r
i
(t
1
)r
∗
j
(t
2
)] =

0 i 6= j
E[|r
i
|
2
]ρ(t
1
−t
2
) i = j
(2)
where ρ(t) denotes the normalized autocorrelation
function of {r
i
(t)} and E(·) denotes expectation.
Consider an N-tone OFDM signal transmitted
over the channel defined by (1). It is assumed that,
when compared with the rate of the OFDM blocks,
the variation of r
i
(t) is slow for all i. Thus, in this
case, the inter-carrier interference (ICI) can be ig-
nored, and the OFDM outputs can be represented by
y
n,k
= h
n,k
x
n,k
+ v
n,k
for all n = 0,1, ··· ,N −1 and
k = ··· , 1, 2,···, where x
n,k
and y
n,k
are, respectively,
the n
th
transmitted and received complex symbols of
the k
th
block. The sequence {h
n,k
}
N−1
n=0
represents the
channel frequency response (CFR) and {v
n,k
}
N−1
n=0
is a
sequence of iid complex Gaussian random variables
with zero mean and fixed variance σ
2
v
for all k and n.
The size of the cyclic prefix (CP) used in this sys-
tem is denoted by L, and the sampling time is T
S
.
Let h(k) = [h
0,k
,··· ,h
N−1,k
]
T
. Then h(k) =
F
L
g(k), where F
L
is the first L columns of the N-point
DFT transform matrix and g(k) = [g
0,k
,··· ,g
L−1,k
]
T
is the discrete time channel impulse response (CIR)
for block k. We have
g
l,k
=
D−1
∑
i=0
r
i
(kT
B
)p(lT
S
−τ
i
), ∀l = 0,1,··· , L−1,
(3)
where p(τ) denote the composite impulse response of
the analog components in the OFDM system, T
S
de-
notes the sampling time, and where T
B
= (N + L)T
S
.
It is inferred from (3) that, although r
i
(t) are uncorre-
lated, the elements of g(k) are correlated. Therefore,
in order to achieve optimal channel prediction, all el-
ements of the outdated CIRs should be used to predict
each element of the current CIR.
2.2 Motivation
In this section we illustrate the effect of outdated
channel state information and motivate the need for
channel prediction. For the purpose of resource allo-
cation, a straightforward approach is to take the most
recent estimate of CSI from the receiver, which is not
only noisy but also outdated, as the predicted value of
the current CSI, i.e., In other words, let
˜
g(k) :=
ˆ
g(k−d) = g(k−d) + e(k −d),
where
ˆ
g(k−d) is the estimated CIR at time k−d,
˜
g(k)
is our prediction of CIR at time k and dT
B
is the as-
sociated delay and e(k −d) is the channel estimation
error.
The normalized mean square error (NMSE), de-
fined by NMSE := E(k
˜
g(k) −g(k)k
2
)/E(kg(k)k
2
), is
used to measure the difference between g(k) and
˜
g(k),
where k·k is the L
2
-norm. It is easy to show that
NMSE = NMSE
˜
g(k)
. Using the WSS assumption of
the channel, the NMSE of the outdated CIR can be
calculated as follows
NMSE = (4)
2E(kgk
2
) + E(kek
2
) −2Re
h
∑
L−1
l=0
E(g
∗
l,k−d
g
l,k
)
i
E(kgk
2
)
Using (2) it can be shown that
E[g
∗
l,k−d
g
l,k
] =
D−1
∑
i=0
D−1
∑
j=0
E[r
∗
i
((k−d)T
B
)r
j
(kT
B
)p
∗
(lT
S
−τ
i
)p(lT
S
−τ
j
)
= ρ(dT
B
)
D−1
∑
i=0
E[|r
i
|
2
]
|
p(lT
S
−τ
i
)
|
2
= ρ(dT
B
)E[g
∗
l,k
g
l,k
] (5)
Using (5) and (5) we can write
NMSE = 2[1−Re(ρ(dT
B
))] + E(kek
2
)/E(kgk
2
)
(6)
Equation (6) shows that if the outdated channel esti-
mate is used as a prediction of the current CSI, the
associated NMSE is not only determined by the sig-
nal to noise ratio (SNR) of the channel estimation
method, but also the autocorrelation function ρ(·). It
turns out that in this case the effect of the channel au-
tocorrelation function is more significant than that of
the estimation error.
For Rayleigh fading channels, ρ(t) = J
0
(2πf
m
t)
(Stuber, 2000), where J
0
(·) is the zeroth-order Bessel
function of the first kind and f
m
denotes the maxi-
mum Doppler frequency shift. Another commonly
used autocorrelation function is ρ(t) = e
−λf
m
t
. For
WINSYS 2007 - International Conference on Wireless Information Networks and Systems
118

λ ≈ 2.8634 this model has the same coherence time
1
as the Rayleigh fading model. The NMSEs calculated
from (6) using these two correlation functions have
been plotted with respect to f
m
dT
B
in Figure 1, where
the SNR in the CIR estimation, namely the value of
E(kgk
2
)/E(kek
2
), is fixed to be 25dB. This figure il-
lustrates that, when a delayed version of the estimated
CIR is used for future CIR prediction, a small delay
may result in large errors in CIR prediction even in
cases where the estimation error is at an acceptable
level (SNR=25dB). For comparison, in the same fig-
ure, we show the results of the CIR prediction using
different predictors, which will be discussed in the
next section.
3 CHANNEL PREDICTION
Channel prediction can be performed for either CFR
or CIR. However, given the fact that the length of the
CFR sequence (N) is often much larger than that of
CIR (L), complexity of the predictor will be signifi-
cantly lower for CIR prediction.
Due to the limited capacity of the feedback
channel, it is assumed that only a down-sampled
version (by a ratio of d > 1) of the CIR esti-
mates at the receiver are fed back to the transmit-
ter. Let
ˆ
g(k −d),
ˆ
g(k −2d), ··· ,
ˆ
g(k −Md) be the
CIR estimations available to the transmitter. The
objective is to find an optimal prediction of g(k)
based on these outdated samples. We rearrange
the elements of these vectors into a single LM-by-
1 vector u(k) := [u
T
0,k
,··· ,u
T
L−1,k
]
T
, where u
l,k
=
[ ˆg
l,(k−d)
,··· , ˆg
l,(k−Md)
]
T
. Then the linear minimum
mean-squared error (MSE) prediction of g(k) is given
by the Wiener-Hopf equation as follows.
˜
g
opt
(k) =

R
−1
uu
P
ug

H
u(k) (7)
where R
uu
= E[u(k)u(k)
H
], P
ug
= E[u(k)g
H
(k)],
and (·)
H
and (·)
−1
denote, respectively, the ma-
trix transpose conjugate and matrix inversion opera-
tions (Haykin, 2002). The definition of u(k) implies
that R
uu
= [A(l
1
,l
2
)]
0≤l
1
,l
2
≤L−1
, where A(l
1
,l
2
) =
E[u
l
1
,k
u
H
l
2
,k
] is a square matrix. Similarly, P
ug
=
[b(l
1
,l
2
)]
0≤l
1
,l
2
≤L−1
, in which b(l
1
,l
2
) = E[u
l
1
,k
g
∗
l
2
,k
]
is a column vector. The MSE associated with this pre-
dictor is given by
NMSE
opt
= 1−Tr(P
H
ug
R
−1
uu
P
ug
)/E[kgk
2
] (8)
where Tr(A) is the trace of the matrix A.
1
The time over which the correlation coefficient is above
0.5
In many cases, when l
1
6= l
2
, the cross-correlation
between g
l
1
,k
1
and g
l
2
,k
2
is relatively small and can
be ignored. Thus, assuming E(g
l
1
,k
1
g
∗
l
2
,k
2
) = 0 for
l
1
6= l
2
, the matrices R
uu
and P
ug
can be rewritten as
R
uu
= diag
{
A(0,0), ··· ,A(L−1, L−1)
}
and P
ug
=
diag
{
b(0,0), ··· ,b(L−1, L−1)
}
. Thus, (7) and (8)
can be simplified as follows.
˜
g
sub
(k) =

b
H
(l, l)A
−1
(l, l)u
l,k

L−1
l=0
(9)
NMSE
sub
= 1−
L−1
∑
l=0
b
H
(l, l)A
−1
(l, l)b(l,l)/E[kgk
2
]
(10)
The structure in (9) is computationally much more ef-
ficient than that in (7) and will be adopted for the re-
mainder of this paper.
3.1 Adaptive Channel Prediction
Using the L predictors in (9) requires the matrices
{A(l, l)}
L−1
l=0
and the vectors {b(l,l)}
L−1
l=0
. However,
for a time-varying channel, frequent computation of
these will be unrealistic. In this case, a more realistic
approach is to use an adaptive prediction scheme such
as least mean-square (LMS), recursive least-squares
(RLS) or a Kalman filter, to replace each of the L
Wiener filters in (9). In this case, the filter coefficients
can be computed recursively. For the LMS predictor,
the processing of the l
th
branch of the predictor can
be represented by
˜g
l,k
= w
H
l,k
u
l,k
(11)
w
l,k+d
= w
l,k
+ ν

ˆg
l,k−d
−w
H
l,k
u
l,k−d

∗
u
l,k−d
(12)
where w
l,k
denotes the filter coefficients of the l
th
branch at time t = kT
B
, and ν is a positive constant
denoting the step size. The choice of ν affects the
convergence properties and the performance of the
predictor and this has been thoroughly discussed in
(Haykin, 2002).
Figure 1 also shows the performance of the opti-
mal and suboptimal Wiener predictors ((7), (9)) and
the LMS predictor ((11)-(12)) in terms of NMSE as-
suming a 12-ray channel model following (1), D = 12.
The path delays {τ
i
}, which are in the interval [0,5]
µsec, and the power gain for each path are provided in
Table 2.1 of (Stuber, 2000). All paths are assumed to
undergo Rayleigh fading and have the normalized au-
tocorrelation function ρ(t) = J
0
(2πf
m
t). The OFDM
parameters used are N = 64, L = 16, T
S
= 0.625µs
(T
B
= 50µs), and p(τ) is set to be the raised-cosine
function with a roll-off factor of 0.35. The SNR of
CIR estimations is set to 25dB, as in Section 2.2. In
this case, d is fixed to be 10 and f
m
varies between 0
and 240Hz. It is clear that, the two Wiener predictors
RESOURCE ALLOCATION FOR OFDM SYSTEMS IN THE PRESENCE OF TIME-VARYING CHANNELS
119

have very close performance and are better than the
LMS predictor. Moreover, even the LMS predictor
shows significant improvement over that of using the
outdated CSI.
4 BIT AND POWER
ALLOCATION
In this section we consider the problem of optimal bit
and power allocation for OFDM systems using the
predicted values of CIR. Since the discussion is fo-
cused on a single block, the subscript indicating the
block number is dropped. Let P
n
and β
n
, respec-
tively, denote the power and the number of bits allo-
cated to subcarrier n. The objective is to minimize the
power allocated to the OFDM block while satisfying
the BER (ε
target
) and data rate (R
target
bits per block)
requirements. In this section the prediction error is
treated as additive noise and is measured by NMSE.
4.1 Resource Allocation with Gaussian
Prediction Error
We assume that the CIR prediction error e = (
˜
g−g) is
a complex-valued Gaussian random vector such that
E[e] = 0
L×1
and E[ee
H
] = σ
2
e
I
L×L
. This assumption
is justified in light of the fact that the predictor is lin-
ear and that all fading components of the channel are
assumed to follow a Gaussian distribution. In (Gao
and Naraghi-Pour, 2006), we discussed the problem
of bit and power allocation for the case of perfect CSI
(i.e., σ
2
e
= 0). In this section, this is extended to the
case of σ
2
e
6= 0.
The instantaneous bit error probability for subcar-
rier n can be written as
BER
n
= c
1
exp

−P
n
|h
n
|
2
q(β
n
)

(13)
where q(·) is a known function of β
n
. It should be
noted that the BER in (13) is evaluated using the chan-
nel CFR {h
n
}, whereas the bit and power allocation is
performed using the predicted values of CFR, namely
{
˜
h
n
}, which in turn is obtained from an N-point DFT
of CIR predictions {˜g
n
}.
For σ
2
e
6= 0, h
n
and thus BER
n
are random vari-
ables. In this case, a constraint regarding system BER
requirement is proposed as follows:
E(BER
n
|
˜
h
n
) = ε
target
, ∀n = 0,··· , N −1. (14)
Using the assumption on channel prediction error be-
ing Gaussian, it can be shown that, given
˜
h
n
, h
n
is a
complex-valued Gaussian random variable with mean
˜
h
n
and variance Lσ
2
e
. Using (13), (14) can be rewritten
as:
c
1
1+ Lξ
n
σ
2
e
exp

−
|
˜
h
n
|
2
ξ
n
1+ Lξ
n
σ
2
e

= ε
target
, (15)
n = 0, 1,··· , N −1,
where ξ
n
= q(β
n
)P
n
. The left hand side of (15) is
monotone decreasing in ξ
n
. Thus for a given ε
target
,
there exists a unique ξ
∗
n
satisfying (15). Let |h
†
n
|
2
:=
ln(
c
1
ε
target
)/ξ
∗
n
. Then P
n
and β
n
satisfy (15) if and only
if
c
1
exp[−P
n
|h
†
n
|
2
q(β
n
)] = ε
target
.
Consequently |h
†
n
|
2
can be defined as the effective
power gain of subcarrier n, and can be calculated as
follows:
|h
†
n
|
2
=
L
2
σ
4
e
ln

c
1
ε
target

/


|
˜
h
n
|
2
Ψ
−1
n
ε
target
c
1
Ψ[
|
˜
h
n
|
2
Lσ
2
e
]
o
−Lσ
2
e


(16)
where Ψ(x) = xe
x
and Ψ
−1
(·) is the inverse function
of Ψ(x).
By treating |h
†
n
|
2
as the perfect power gain for sub-
channel n, the bit and power allocation problem can
be solved using known algorithms developed for the
case of perfect CSI (Gao and Naraghi-Pour, 2006;
Krongold et al., 2000). However, unlike the approach
of directly using |
˜
h
n
|
2
, the proposed method guaran-
tees that the BER satisfies the requirement in (14).
4.2 Resource Allocation with Arbitrary
Channel Prediction Error
Our discussion in the previous section assumed that
the values of CFR, {h
n
}, are predicted. However, in
bit and power allocation algorithm it is {|h
n
|
2
}, the
channel power gain, (or more specifically the SNR of
each subcarrier) that is required. As argued in (Fala-
hati et al., 2004), using the square magnitude of
˜
h
n
as a prediction of the channel power gain underes-
timates the true power and results in a biased esti-
mate. Consequently in (Falahati et al., 2004), an unbi-
ased quadratic power prediction method from (Ekman
et al., 2002) has been used for optimal rate and power
allocation.
In this section we consider the problem of opti-
mal bit and power allocation in OFDM assuming an
arbitrary distribution for the channel CFR. In particu-
lar let X
n
:= |h
n
|
2
and Y
n
:= |
˜
h
n
|
2
. It is assumed that
the joint PDF of X
n
and Y
n
, denoted by f
X
n
,Y
n
(x,y), is
known, whether
˜
h
n
or |
˜
h
n
|
2
is obtained through chan-
nel prediction. For all n, both X
n
and Y
n
are viewed
WINSYS 2007 - International Conference on Wireless Information Networks and Systems
120

as complex random variables. As before, P
n
and β
n
are, respectively, the power and the number of bits al-
located to the n
th
subchannel.
In the optimal bit and power allocation P
n
and β
n
are determined by the value of Y
n
. Since BER also de-
pends on the value of X
n
, (see (13)), as in the previous
section the BER constraint is considered as follows.
E (BER
n
|Y
n
= y) =
∞
0
c
1
e
−q(β
n
)P
n
x
f
X
n
|Y
n
(x;y)dx
= ε
target
, ∀n = 1,··· , N. (17)
Define
Z
n
(z;y) =
∞
0
e
−zx
f
X
n
|Y
n
(x;y)dx
Then, for each y, Z
n
(z;y) is a monotone decreasing
function of z. Let Z
−1
n
(·;y) denote its inverse func-
tion such that Z
n
(Z
−1
n
(x;y);y) ≡ x. Thus (17) can be
rewritten as
P
n
=
1
q(β
n
)
Z
−1
n

ε
target
c
1
;y

(18)
Let Ω = {b
0
,b
1
,··· ,b
M
} be the set of integers
that β
n
can assume. Divide the interval [0, ∞) into
M consecutive subintervals with the boundary points
0 = ϕ
0,n
< ϕ
1,n
< ··· < ϕ
M ,n
< ϕ
M +1,n
= ∞. Then,
let β
n
= b
m
if the value of Y
n
falls in the interval
(ϕ
m,n
,ϕ
m+1,n
]. Finally, calculate P
n
from (18) with
β
n
= b
m
. The same procedure will be performed for
all n to obtain resource allocation for the entire block.
From (18), the transmit power of the OFDM block is
given by
¯
P =
N
∑
n=1
∞
0
1
q(β
n
)
Z
−1
n

ε
target
c
1
;y

f
Y
n
(y)dy
=
N
∑
n=1
M
∑
m=1
1
q(b
m
)
ϕ
m+1,n
ϕ
m,n
Z
−1
n

ε
target
c
1
;y

f
Y
n
(y)dy,
(19)
and the data rate is given by
¯
R =
N
∑
n=1
M
∑
m=1
b
m
ϕ
m+1,n
ϕ
m,n
f
Y
n
(y)dy = R
target
(20)
As mentioned previously, the bit and power alloca-
tion algorithm attempts to minimize the total power
assigned to an OFDM block subject to the constraints
on the BER and the data rate per OFDM block. It
can be shown that there exist optimal boundary val-
ues {ϕ
∗
m,n
} such that the transmit power in (19) can
be minimized subject to (17) and rate constraint in
¯
R = R
total
. This problem can be solved using the
method of Lagrange multipliers.
The Lagrange cost function is
J(ϕ
1,1
,··· ,ϕ
M ,N
)
=
N
∑
n=1
M
∑
m=1
1
q(b
m
)
ϕ
m+1,n
ϕ
m,n
Z
−1
n

ε
target
c
1
;y

f
Y
n
(y)dy
+Λ
"
N
∑
n=1
M
∑
m=1
b
m
ϕ
m+1,n
ϕ
m,n
f
Y
n
(y)dy−R
target
#
(21)
where Λ is the Lagrange multiplier. The necessary
conditions for optimality are given by
∂J
∂ϕ
m,n




ϕ
m,n
=ϕ
∗
m,n
= 0, ∀m, n (22)
which yield
Z
−1
n

ε
target
c
1
;ϕ
∗
m,n

= −
Λ(b
m
−b
m−1
)
[
1
q(b
m
)
−
1
q(b
m−1
)
]
(23)
for n = 1, ··· ,N; m = 1,··· ,
M .
For a given value of Λ, we can obtain {ϕ
∗
m,n
} by solv-
ing the above equations. For a given set of thresholds
{ϕ
∗
m,n
} we can obtain the value of Λ, from (20). In
practice, a recursive numerical method can be used to
solve for {ϕ
∗
m,n
} and Λ.
Example 4.1 Suppose that the prediction of channel
power gain is perfect, i.e., Y
n
= X
n
for all n. In this
case, f
X
n
|Y
n
(x;y) = δ(x−y) and Z
n
(z;y) = e
−zy
. Thus
(23) can be reduced to
ϕ
∗
m,n
=
ln(
ε
target
c
1
)
Λ
"
1
q(b
m
)
−
1
q(b
m−1
)
b
m
−b
m−1
#
, ∀m, n. (24)
Equation (24) shows that, for each subcarrier, the ra-
tio of the optimal threshold values are fixed. A re-
sult which coincides with earlier results in (Gao and
Naraghi-Pour, 2006; Krongold et al., 2000). More-
over, the values given by (24) are the same as those
obtained in (Krongold et al., 2000), where these opti-
mal threshold values are derived using a different ap-
proach.
Example 4.2 Suppose
˜
h
n
= h
n
+η
n
, where h
n
and η
n
are independent complex Gaussian random variables
satisfying h
n
∼
C N (0,θ
2
) and η
n
∼ C N (0,σ
2
η
),
where θ
2
and σ
2
η
are given. When conditioned on
˜
h
n
, h
n
is a complex-valued Gaussian random variable
with mean
˜
h
n
and variance σ
2
η
. It is clear that given
Y
n
= |
˜
h
n
|
2
, X
n
= |h
n
|
2
has a non-central chi-square
distribution with two degrees of freedom. Accord-
ingly, f
X
n
|Y
n
(x;y) can be written as
f
X
n
|Y
n
(x;y) =
1
σ
2
η
exp −
x+ y
σ
2
η
!
I
0
√
xy
σ
2
η
/2
!
(25)
where I
0
(·) denotes the zeroth order modified Bessel
function of the first kind. For the conditional distribu-
tion defined in (25), the function Z
n
can be rewritten
RESOURCE ALLOCATION FOR OFDM SYSTEMS IN THE PRESENCE OF TIME-VARYING CHANNELS
121

as follows:
Z
n
(z;y) =
exp

−(
yz
1+σ
2
η
z
)

1+ σ
2
η
z
(26)
In this case, (23) can be simplified as
ϕ
∗
m,n
=

1
Λ∆
m
+ σ
2
η

ln
c
1
/ε
target
1+ Λσ
2
η
∆
m
!
(27)
for all n = 1, ··· ,N; m = 1,··· ,
M .
where ∆
m
= −(b
m
−b
m−1
)/(
1
q(b
m
)
−
1
q
(
b
m−1
)
). It is ob-
served that for σ
2
η
= 0, (27) reduces to (24). In gen-
eral, the Lagrange multiplier Λ is determined by the
constraint in (20). Specifically, if σ
2
η
and θ
2
are iden-
tical for all n, then (20) can be rewritten as follows:
M
∑
m=1
b
m
θ
2

e
−ϕ
∗
m
/θ
2
−e
−ϕ
∗
m+1
/θ
2

=
R
target
N
(28)
5 SIMULATION RESULTS
In this section, we first illustrate through simulation
the efficacy of the resource allocation schemes pro-
posed in Section 4. Meanwhile, the 12-ray chan-
nel model described in Section 3.1 has been used in
this simulation, and the value of f
m
is set to 240Hz.
The other system parameters are the same as those
in Section 3.1, in which d = 10. The set of modula-
tion schemes used in this case are QPSK, 16QAM,
64QAM and 256QAM, and the target data rate is
R
target
= 4N. From Figure 1, the NMSE of the LMS
predictor for f
m
= 240Hz is about −15dB. Thus, for
simplicity, we ignore the design of the channel pre-
dictor by assuming its output is a (Gaussian) noisy
version of the CIR with NMSE=−15dB. In other
words, the predicted CIR used for resource allocation
is generated by adding to g(k) (which is calculated
using (3)), a sequence of iid complex-valued Gaus-
sian random variables, whose variance is determined
by NMSE.
The allocation methods proposed in Sections 4.1
and 4.2 are compared with the results in (Gao and
Naraghi-Pour, 2006) (which assumes perfect knowl-
edge of the CSI). We should point out that if knowl-
edge of the CSI is perfect, then the method in (Gao
and Naraghi-Pour, 2006) is optimal. For these three
approaches, the BER values measured from simula-
tion are plotted vs. the target BER values in Figure
2. It is clear that both methods in Section 4 meet
the BER requirement while the approach in (Gao and
Naraghi-Pour, 2006) does not.
The spectral efficiency of the above three alloca-
tion methods is compared in Figure 3, where the re-
sults corresponding to the perfect CSI case (NMSE=
−∞ dB) is also plotted. For the method in Section 4.1,
since, in the case of perfect CSI, the effective power
gain |h
†
n
|
2
in (16) equals the predicted value |
˜
h
n
|
2
, then
the scheme in Section 4.1 is equivalent to that in (Gao
and Naraghi-Pour, 2006). Thus, only two curves ex-
ist in the case of perfect CSI. In the case of perfect
CSI, the approach in Section 4.2 outperforms that in
Section 4.1. This can be explained as follows: the
method in Section 4.1 requires each OFDM block to
transmit R
target
bits, while the method in Section 4.2
has a more relaxed condition (20) and should result in
higher efficiency. However, it should be noted that the
scheme in Section 4.1 is easier to implement. For the
case of imperfect CSI (NMSE=−15dB), the resource
allocation scheme in Section 4.1 has about 2.5dB im-
provement over the method in (Gao and Naraghi-
Pour, 2006) for ε
target
= 10
−4
, and the method in Sec-
tion 4.2 is about 3.5dB better than that in (Gao and
Naraghi-Pour, 2006).
We also simulated a complete OFDM system with
both channel prediction and resource allocation. The
same channel model and system parameters as those
in previous simulations have been used in this system,
where the range of f
m
is 0−150Hz and R
target
= 2N.
For every 20 OFDM blocks, one channel estimation
is sent back to the transmitter, i.e., d = 20 in this case.
The channel predictor used here is the LMS predictor
in (11)-(12) with M=5.
Figure 4 illustrates the simulation results. A ref-
erence system without channel prediction, which uses
the most recent channel estimation has also been sim-
ulated. The results of this case are labeled as “non-
pred.” in this figure. The system having channel pre-
diction and the proposed resource allocation clearly
outperforms the reference system under all Doppler
frequencies. In this case, the efficiency of the pro-
posed system is almost as good as that of perfect CSI
for Doppler frequencies up to 100Hz. On the other
hand, the reference system suffers tremendous perfor-
mance loss even for f
m
= 50Hz. For ε
target
= 10
−3
,
the proposed system results in 2dB improvement over
the reference system for f
m
= 50Hz, 8dB improve-
ment for f
m
= 100Hz, and 10dB improvement for
f
m
= 150Hz.
WINSYS 2007 - International Conference on Wireless Information Networks and Systems
122

6 CONCLUSION
The problem of resource allocation with imperfect
CSI has been considered for OFDM systems and
time-varying channels. The outdated CSI is identi-
fied as the main source of difficulty in achieving the
performance enhancement promised by resource allo-
cation techniques. With the aid of channel prediction,
a bit and power allocation scheme has been proposed
to overcome this difficulty. The simulation results
confirm that, using the proposed method, the system
performance for slowly time-varying channels (e.g.,
f
m
≤ 100Hz in our simulation) can be very close to
that of time-invariant channels.
0 0.02 0.04 0.06 0.08 0.1 0.12
−40
−35
−30
−25
−20
−15
−10
−5
0
f
m
⋅(dT
B
)
NMSE (dB)
ρ(t)=J
0
(2π f
m
t), no channel prediction
ρ(t)=exp(−λ f
m
t), no channel prediction
ρ(t)=J
0
(2π f
m
t), LMS predictor (11)−(12)
ρ(t)=J
0
(2π f
m
t), Wiener predictor (9)
ρ(t)=J
0
(2π f
m
t), Wiener predictor (7)
Figure 1: Performance of channel prediction in terms of
NMSE.
10
−5
10
−4
10
−3
10
−2
10
−5
10
−4
10
−3
10
−2
Target BER (ε
target
)
BER
Method in Ref. [4], NMSE=−15dB
Method in Sec. 4.1, NMSE=−15dB
Method in Sec. 4.2, NMSE=−15dB
Figure 2: Comparison between the measured BER and the
target BER.
16 18 20 22 24 26 28 30
10
−5
10
−4
10
−3
10
−2
SNR (dB)
BER
Perfect CSI, Method in Sec. 4.2
Perfect CSI, Method in Ref. [4]
NMSE=−15dB, Method in Sec 4.2
NMSE=−15dB, Method in Sec. 4.1
NMSE=−15dB, Method in Ref. [4]
Figure 3: Performance of resource allocation schemes for
imperfect CSI.
5 10 15 20 25 30
10
−5
10
−4
10
−3
10
−2
SNR (dB)
BER
f
m
=0Hz (perfect CSI)
f
m
=50Hz, pred.
f
m
=100Hz, pred.
f
m
=150Hz, pred.
f
m
=50Hz, non−pred.
f
m
=100Hz, non−pred.
f
m
=150Hz, non−pred.
Figure 4: A comparison of predictive and non-predictive
resource allocation schemes.
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